Question

Find the middle term of an A.P. $20,16,12,....., - 176$.

Answer 1

Hint: An arithmetic sequence or arithmetic progression is defined as a mathematical sequence in which the difference between two consecutive terms is always a constant. An arithmetic progression is abbreviated as A.P. We also need to learn the three important terms, which are as follows.
A common difference $\left( d \right)$ is the difference between the first two terms.
${n^{th}}$term \[({a_n})\]
And, Sum of the first $n$ terms \[({S_n})\]
Here, we are asked to calculate the middle term of the given arithmetic progression.
And the given arithmetic progression is $20,16,12,....., - 176$
First, we need to use the formula to calculate the last term of the arithmetic progression so that we can calculate the number of terms in the given A.P. If the number of terms is calculated, it will be easy to obtain the required middle terms.
Formula used:
The formula to calculate the ${n^{th}}$term of the given arithmetic progression is as follows.
\[\begin{array}{*{20}{l}}
  {{{\mathbf{a}}_{\mathbf{n}}} = {\mathbf{a}} + \left( {{\mathbf{n}} - {\mathbf{1}}} \right) \times {\mathbf{d}}}
\end{array}\]
Where, $a$ denotes the first term, $d$ denotes the common difference,$n$ is the number of terms, and ${a_n}$ is the ${n^{th}}$ term of the given arithmetic progression.
If the number of terms is even, then the middle term will be found using the formula
Middle term $ = {\left( {\dfrac{n}{2}} \right)^{th}}and{\left( {\dfrac{n}{2} + 1} \right)^{th}}terms$

Complete step by step answer:
The given arithmetic progression is $20,16,12,....., - 176$
Here, the first term is
$a = 20$
Also, a common difference is $d = 16 - 20$
$d = - 4$
Let us calculate the number of terms of the given A.P using the formula.
\[\begin{array}{*{20}{l}}
  {{{\mathbf{a}}_{\mathbf{n}}} = {\mathbf{a}} + \left( {{\mathbf{n}} - {\mathbf{1}}} \right) \times {\mathbf{d}}}
\end{array}\]
      \[ - 176\begin{array}{*{20}{l}}
  { = 20 + \left( {n - 1} \right) \times - 4}
\end{array}\]
\[ \Rightarrow - 176\begin{array}{*{20}{l}}
  { = 20 - 4}
\end{array}n + 4\]
\[ \Rightarrow - 176\begin{array}{*{20}{l}}
  { = 24 - 4}
\end{array}n\]
\[ \Rightarrow 4n = 176 + 24\]
\[ \Rightarrow 4n = 200\]
\[ \Rightarrow n = \dfrac{{200}}{4}\]
\[ \Rightarrow n = 50\]
Hence, there are a total fifty ($50$ ) terms in the given A.P .
If the number of terms is even, then the middle term will be found using the formula
Middle term $ = {\left( {\dfrac{n}{2}} \right)^{th}}and{\left( {\dfrac{n}{2} + 1} \right)^{th}}terms$
Since the number of terms in the given A.P is even, the middle terms will be ${\left( {\dfrac{{50}}{2}} \right)^{th}}term$ and${\left( {\dfrac{{50}}{2} + 1} \right)^{th}}term$
Hence, the middle terms are ${25^{th}}$ term and ${26^{th}}$ term.
Now, we need to substitute $n = 25$ and $n = 26$ to obtain the middle terms.
\[\begin{array}{*{20}{l}}
  {{{\mathbf{a}}_{\mathbf{n}}} = {\mathbf{a}} + \left( {{\mathbf{n}} - {\mathbf{1}}} \right) \times {\mathbf{d}}}
\end{array}\]
a) $n = 25$
\[\begin{array}{*{20}{l}}
  {{a_n} = 20 + \left( {25 - 1} \right) \times - 4}
\end{array}\]
\[\begin{array}{*{20}{l}}
  { \Rightarrow {a_n} = 20 - 100 + 4}
\end{array}\]
\[\begin{array}{*{20}{l}}
  { \Rightarrow {a_n} = - 76}
\end{array}\]
b) $n = 26$
\[\begin{array}{*{20}{l}}
  {{a_n} = 20 + \left( {26 - 1} \right) \times - 4}
\end{array}\]
\[\begin{array}{*{20}{l}}
  { \Rightarrow {a_n} = 20 - 100}
\end{array}\]
\[\begin{array}{*{20}{l}}
  { \Rightarrow {a_n} = - 80}
\end{array}\]
Therefore, the required middle terms are $ - 76$ and $- 80$

Note: We won’t forget the three important terms while studying A.P, which are as follows.
A common difference$\left( d \right)$ is the difference between the first two terms.
${n^{th}}$term\[({a_n})\]
And, Sum of the first $n$ terms\[({S_n})\]
If the number of terms is even, then the middle term will be found using the formula
Middle term $ = {\left( {\dfrac{n}{2}} \right)^{th}}term$